Fuzzy Logic and Fuzzy Systems

Transcription

1 Fuzzy Logic and Fuzzy Systems Revision Lecture Khurshid Ahmad, Professor of Computer Science, Department of Computer Science Trinity College, Dublin-2, IRELAND 24 February

2 Knowledge Representation & Reasoning We have covered five topics in this course: 1. Terminology: Uncertainty, Approximations and Vagueness 2. Fuzzy Sets 3. Fuzzy Logic and Fuzzy Systems 4. Fuzzy Control 5. Neuro-fuzzy systems 2 2

3 The Written Examination 3 3

4 Each question has a preamble that defines the scope of the question. The Written Examination There is a clear indication as to which of the five topics the question covers. 4 4

5 Each question has two component: A conceptual part testing your comprehension of terminology and ontology of the subject carrying no more than 1/3 of the total mark for the question The Written Examination 5 5

6 Each question has two component: A conceptual part testing your comprehension of terminology and ontology of the subject And a problem to be solved which shows your ability to deploy your knowledge. This part comprises 2/3 of the mark for the question The Written Examination 6 6

7 Each question has a preamble that defines the scope of the question. The Written Examination There is a clear indication as to which of the five topics the question covers. 7 7

8 Each question has two component: A conceptual part testing your comprehension of terminology and ontology of the subject carrying no more than 1/3 of the total mark for the question The Written Examination 8 8

9 Each question has two component: A conceptual part testing your comprehension of terminology and ontology of the subject And a problem to be solved which shows your ability to deploy your knowledge. This part comprises 2/3 of the mark for the question The Written Examination 9 9

10 The Written Examination I will prefer your answer to the conceptual part should be short and scuccinct

11 The Written Examination For the problemsolving part, please be sure to show how you performed the calculation. Comment on the steps you have taken

12 The Written Examination For the problemsolving part, please be sure to show how you performed the calculation. Comment on the steps you have taken

13 The Written Examination For the problemsolving part, please be sure to show how you performed the calculation. Comment on the steps you have taken

14 UNCERTAINITY AND ITS TREATMENT Theory of fuzzy sets and fuzzy logic has been applied to problems in a variety of fields: Taxonomy; Topology; Linguistics; Logic; Automata Theory; Game Theory; Pattern Recognition; Medicine; Law; Decision Support; Information Retrieval; And more recently FUZZY Machines have been developed including automatic train control and tunnel digging machinery to washing machines, rice cookers, vacuum cleaners and air conditioners

15 UNCERTAINITY AND ITS TREATMENT The term fuzzy logic is used in two senses: Narrow sense: Fuzzy logic is a branch of fuzzy set theory, which deals (as logical systems do) with the representation and inference from knowledge. Fuzzy logic, unlike other logical systems, deals with imprecise or uncertain knowledge. In this narrow, and perhaps correct sense, fuzzy logic is just one of the branches of fuzzy set theory. Broad Sense: fuzzy logic synonymously with fuzzy set theory 15 15

16 FUZZY SETS An Example: Consider a set of numbers: X = {1, 2,.. 10}. Johnny s understanding of numbers is limited to 10, when asked he suggested the following. Sitting next to Johnny was a fuzzy logician noting : Large Number , 4, 3, 2, 1 Comment Surely Surely Quite poss. Maybe In some cases, not usually Definitely Not Degree of membership

17 FUZZY SETS An Example: Consider a set of numbers: X = {1, 2,.. 10}. Johnny s understanding of numbers is limited to 10, when asked he suggested the following. Sitting next to Johnny was a fuzzy logician noting : Large Number , 4, 3, 2, 1 Comment Surely Surely Quite poss. Maybe In some cases, not usually Definitely Not Degree of membership We can denote Johnny s notion of large number by the fuzzy set A =0/1+0/2+0/3+0/4+0/5+ 0.2/ / /8 + 1/9 + 1/

18 FUZZY SETS Fuzzy (sub-)sets: Membership Functions For the sake of convenience, usually a fuzzy set is denoted as: A = µ A (x i )/x i +. + µ A (x n )/x n that belongs to a finite universe of discourse: A x, x,..., } ~ { 1 2 x n whereµ A (x i )/x i (a singleton) is a pair grade of membership element

19 FUZZY SETS: PROPERTIES Properties P 1 P 2 P 3 P 4 P 5 Equality of two fuzzy sets Inclusion of one set into another fuzzy set Cardinality of a fuzzy set An empty fuzzy set α-cuts Definition 19 19

20 FUZZY SETS: OPERATIONS Operations O 1 Definition & Example The complementation of a fuzzy set A X (A of X) A (NOT A of X) ~ µ A (x) = 1 - µ A (x) Example: Recall X = {1, 2, 3} and A = 0.3/ /2 + 1/3 A = A = 0.7/ /2. Example: Consider Y = {1, 2, 3, 4} and C Y ~ C = 0.6/ /2 + 1/3; then C = ( C) = 0.4/ /2 + 1/4 then C = ( C) = 0.4/ /2 + 1/4; C 1 contains one member not in C (i.e., 4) and does not contain one member of C (i.e., 3) 20 20

21 Knowledge Representation & Reasoning Once we have found that the knowledge of a specialism can be expressed through linguistic variables and rules of thumb, that involve imprecise antecedents and consequents, then we have a basis of a knowledge-base. In this knowledge-base facts are represented through linguistic variables and the rules follow fuzzy logic. In traditional expert systems facts are stated crisply and rules follow classical propositional logic

22 Knowledge Representation & Reasoning A fuzzy knowledge-based system (KBS) is a KBS that performs approximate reasoning. Typically a fuzzy KBS uses knowledge representation and reasoning in systems that are based on the application of Fuzzy Set Theory. A fuzzy knowledge base comprises vague facts and vague rules of the form: KB Entity Fact Rule Fuzzy KB X is µ X IF X is µ X THEN Y is µ Y Crisp KB X is TRUE or X is NOT TRUE IF X THEN Y 22 22

23 Knowledge Representation There are two challenges: (a)how to interpret and how to represent vague rules with the help of appropriate fuzzy sets? & (b)how to find an inference mechanism that is founded on well-defined semantics and that permits approximate reasoning by means of a conjunctive general system of vague rules and case-specific vague facts? 23 23

24 Knowledge Representation Linguistic Variables A linguistic variable is associated with two rules: (a)a syntactic rule, which defines the wellformed sentences in T( ); and (b)a semantic rule, by which the meaning of the terms in T( ) may be determined. If X is a term in T( ), then its meaning (in a denotational sense) is a subset of U. A primary fuzzy set, that is, a term whose meaning must be defined a priori, and serves as a basis for the computation of the meaning of the nonprimary terms in T( )

25 Knowledge Representation & Reasoning R E C A P I T U L A T E 25 25

26 Knowledge Representation & Reasoning The operation of a fuzzy expert system depends on the execution of FOUR major tasks: Fuzzification, Inference, Composition, Defuzzification

27 Knowledge Representation & Reasoning Fuzzification involves the choice of variables, fuzzy input and output variables and defuzzified output variable(s), definition of membership functions for the input variables and the description of fuzzy rules

28 Knowledge Representation & Reasoning Fuzzification : The membership functions defined on the input variables are applied to their actual values to determine the degree of truth for each rule premise. The degree of truth for a rule's premise is sometimes referred to as its α (alpha) value. If a rule's premise has a non-zero degree of truth, that is if the rule applies at all, then the rule is said to fire

29 Knowledge Representation & Reasoning Inference: The truth-value for the premise of each rule is computed and the conclusion applied to each part of the rule. This results in one fuzzy subset assigned to each output variable for each rule

30 Knowledge Representation & Reasoning Inference: MIN and PRODUCT are two inference methods. 1. In MIN inferencing the output membership function is clipped off at a height corresponding to the computed degree of truth of a rule's premise. This corresponds to the traditional interpretation of the fuzzy logic's AND operation. 2. In PRODUCT inferencing the output membership function is scaled by the premise's computed degree of truth

31 Knowledge Representation & Reasoning Composition: All the fuzzy subsets assigned to each output variable are combined together to form a single fuzzy subset for each output variable

32 Knowledge Representation & Reasoning Composition: MAX and SUM are two composition rules: 1. In MAX composition, the combined fuzzy subset is constructed by taking the pointwise maximum over all the fuzzy subsets assigned to the output variable by the inference rule. 2. The SUM composition, the combined output fuzzy subset is constructed by taking the pointwise sum over all the fuzzy subsets assigned to output variable by their inference rule. (Note that this can result in truth values greater than 1)

33 Knowledge Representation & Reasoning Defuzzification: The fuzzy value produced by the composition stage needs to be converted to be converted to a single number or a crisp value

34 Knowledge Representation & Reasoning Defuzzification: The crisp value is essentially the area under the curve of the new fuzzy subset derived from the composition stage. Such a computation takes into account the effect of each rule ina proportionate manner. Sometimes, however, it is important to take only into account those rules that have the maximum impact. Hence there are different methods of defuzzication

35 Knowledge Representation & Reasoning Defuzzification: Two popular defuzzification techniques are the CENTROID and MAXIMUM techniques. 1. The use of CENTROID technique relies on using the centre of gravity of the membership function to calculate the crisp value of the output variable. 2. The MAXIMUM techniques, and there are a number of them, broadly speaking, use one of the variable values at which the fuzzy subset has its maximum truth value to compute the crisp value

36 Knowledge Representation & Reasoning: The Air-conditioner Example DEFUZZIFICATION: The Centre of Gravity (COG) of the output of the rules: Formally, the crisp value is the value located under the centre of gravity of the area that is given by the function η = yε Y µ output x x n ( y) dy yε Y y µ output x x n ( y) dy 36 36

37 Knowledge Representation & Reasoning: The Air-conditioner Example DEFUZZIFICATION: The crisp value h can be obtained by approximating the integral with a sum η = 1 Σy Σµ output ( y) x x n µ output x x n ( y) The centre of gravity approach attempts to take the rules into consideration according to their degree of applicability. If a rule dominates during a certain interval then its dominance is discounted in other intervals

38 Knowledge Representation & Reasoning: The Air-conditioner Example DEFUZZIFICATION: Another method of defuzzification is that of Mean of Maxima (MOM) Method. Here again the weighted sum and weighted membership are worked out, except that the membership function is given another alpha level cut corresponding to the maximum value of the output fuzzy set. The crisp value for MOM method is given as: η = Max 1 ( µ output x ) Σ output. x y= Max ( n x1... xn 1... µ ) y 38 38

39 Knowledge Representation & Reasoning: The Air-conditioner Example What kind of fuzzy logic we have been discussing? Mamdani calculus where membership functions of both antecedant and consequent variables are to be considered at the composition stage. Mamdani calculus involves computation of the consequent fuzzy variables. This is not always possible for real-time systems for example running at high throughput rates- or not always desirable on the basis of Occam s logic; things to be kept simple wherever possible. So if you can approximate a function with a single variable then this is better than having a function; when possible the approximation of a constant is better than having a variable

40 SYSTEMS Knowledge Representation & Reasoning: The Air-conditioner Example Salary Membership Function Debt Membership Function Membership Function Excellent Good Poor Membership Vlaue Small Large Salary in '000 Euros Debt in '000 Euros Risk Membership Functions Membership Value % 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Risk Low Medium High 40 40

41 SYSTEMS Knowledge Representation & Reasoning: The Air-conditioner Example Risk Membership Functions Salary Membership Function Membership Value % 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Risk Debt Membership Function Low Medium High Membership Function Excellent Good Poor Salary in '000 Euros Salary =95K, Debts=60K Original and alpha-cut Membership Functions Membership Vlaue Small Large Membership Value Low Alpha_Low Medium Alpha_Medium High Alpha-High Debt in '000 Euros Risk 41 41

42 SYSTEMS Knowledge Representation & Reasoning: The Air-conditioner Example Risk Membership Functions Salary Membership Function Membership Value % 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Risk Debt Membership Function Low Medium High Membership Function Salary in '000 Euros Salary =50K, Debts=40K Original and alpha-cut Membership Functions Excellent Good Poor Membership Vlaue Small Large Membership Value Low Alpha_Low Medium Alpha_Medium High Alpha-High Debt in '000 Euros Risk 42 42

43 FUZZY CONTROL Control Theory? The term control is generally defined as a mechanism used to guide or regulate the operation of a machine, apparatus or constellations of machines and apparatus

44 FUZZY CONTROL CONTROL THEORY? 'Feedback control' is thus a mechanism for guiding or regulating the operation of a system or subsystems by returning to the input of the (sub)system a fraction of the output

45 FUZZY CONTROL DEFINITIONS 'Feedback control' is thus a mechanism for guiding or regulating the operation of a system or subsystems by returning to the input of the (sub)system a fraction of the output. w e C u S y The machinery or apparatus etc., to be guided or regulated is denoted by S, the input by W and the output by y, and the feedback controller by C. The input to the controller is the socalled error signal e and the purpose of the controller is to guarantee a desired response of the output y

46 FUZZY CONTROL FUZZY CONTROLLERS Here are some heuristics for making decisions in a feedback control loop: System Responsiveness IF the error is positive (negative) & the change in error is approximately zero THEN the change in control is positive (negative); Reduction in overshooting IF the error is approximately zero & the change in error is positive (negative) THEN the change in control is positive (negative); Steady State Control IF the error and change in error are approximately zero THEN the change in control is approximately zero

47 FUZZY CONTROL Balancing the Cartpole The Cartpole Problem is often used to illustrate the use of fuzzy logic. Basically, we have a pole of length l, with a mass m at its head and mass M at its base, has to be kept upright. The application of a force F is required to control the pole. These two masses are connected by a weightless shaft. The base can be moved on a horizontal axis. The angle of the pole in relation to the vertical axis (θ), and the angular velocity (dθ/dt) are two OUTPUT variables Kruse, R., Gebhardt, J., & Klawonn (1994). Foundations of fuzzy systems. Chichester: John Wiley & Sons Ltd 47 47

48 FUZZY CONTROL Control Theory? Typically, rules contain membership functions for both antecedents and consequent. Mamdani Controller If e(k) is positive(e) and e(k) is positive( e) then u(k) is positive ( u) Takagi-Sugeno Controllers: If e(k) is positive(e) and e(k) is positive( e) then u(k) =αe(k)+ß e(k); α and ß are obtained from empirical observations by relating the behaviour of the errors and change in errors over a fixed range of changes in control 48 48

49 FUZZY CONTROL FUZZY CONTROLLERS A fuzzy logic controller (FLC) with a rule base is defined by the matrix: e(k) e(k) N Z P N N N Z Z N Z P P Z P P where the matrix interrelates the error value e(k) in at a given time k, e(k) denotes the change in error (= e(k) - e(k-1)), and the control change u(k) is defined as the difference between u(k) and u(k-1). The term-sets of the input and output variables of the FLC error e, error change e and control change u by the linguistic labels negative (N), approximately zero (Z) and positive I(P). The above FLC matrix can equivalent antecedent/consequent rule set 49 49

50 FUZZY CONTROL FUZZY CONTROLLERS A CONTROL PROCEDURE FIND the firing level of each of the rules FUZZIFICATION FIND the output of each of the rules INFERENCE AGGREGATE the individual rule outputs to obtain the overall system output COMPOSITION OBTAIN a crisp value to be input to the controlled system DEFUZZIFICATION 50 50

51 51 51 FUZZY CONTROL FUZZY CONTROLLERS- An example The membership functions for the three elements of the term set for the error e are given as: = = = = µ + = + = = µ = = = µ ) sgn( 2 & 2 0 ) ( ) ( ) ( e e e e e e e e e e e e e e e e error zero error positive error negative

52 FUZZY CONTROL FUZZY CONTROLLERS Another example For the case where e(k)= -0.9 and e(k)= 0.2, the level or degree of firing for the 9-rule rule set: e & e τ (=min {e, e}) Negative Zero Positive Negative Zero Positive Output Rule & 0 0 Rule & Rule & Rule & 0 0 Rule & Rule & Rule7 0 & 0 0 Rule8 0 & Rule9 0 &

53 FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno Controllers According to Yager and Filev, a known disadvantage of the linguistic modules is that they do not contain in an explicit form the objective knowledge about the system if such knowledge cannot be expressed and/or incorporated into fuzzy set framework' (1994:192). Typically, such knowledge is available often: for example in physical systems this kind of knowledge is available in the form of general conditions imposed on the system through conservation laws, including energy mass or momentum balance, or through limitations imposed on the values of physical constants

54 FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno Controllers Tomohiro Takagi and Michio Sugeno recognised two important points: 1. Complex technological processes may be described in terms of interacting, yet simpler sub processes. This is the mathematical equivalent of fitting a piece-wise linear equation to a complex curve. 2. The output variable(s) of a complex physical system, e.g. complex in the sense it can take a number of input variables to produce one or more output variable, can be related to the system's input variable in a linear manner provided the output space can be subdivided into a number of distinct regions. Takagi, T., & Sugeno, M. (1985). Fuzzy Identification of Systems and its Applications to Modeling and Control. IEEE Transactions on Systems, Man and Cybernetics. Volume No. SMC-15 (No.1) pp

55 FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno Controllers Mamdani style inference: The Bad News: This method involves the computation of a twodimensional shape by summing, or more accurately integrating across a continuously varying function. The computation can be expensive. For every rule we have to find the membership functions for the linguistic variables in the antecedents and the consequents; For every rule we have to compute, during the inference, composition and defuzzification process the membership functions for the consequents; Given the non-linear relationship between the inputs and the output, it is not easy to identify the membership functions for the linguistic variables in the consequent 55 55

56 FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno Controllers Takagi and Sugeno (1985) have argued that in order to develop a generic and simple mathematical tool for computing fuzzy implications one needs to look at a fuzzy partition of fuzzy input space. In each fuzzy subspace a linear inputoutput relation is formed. The output of fuzzy reasoning is given by the values inferred by some implications that were applied to an input

57 FUZZY CONTROL FUZZY CONTROLLERS Takagi-Sugeno Controllers Takagi and Sugeno have described a fuzzy implication R is of the format: R: if (x 1 is µ A (x 1 ), x k is µ A (x k )) then y = g(x 1,, x k ), where: A zero order Takagi-Sugeno Model will be given as R: if (x 1 is µ A (x 1 ), x k is µ A (x k )) then y = k 57 57

58 Knowledge Representation & Reasoning: The Air-conditioner Example Let the temperature be 5 degrees centigrade: Fuzzification: 5 degrees means that it can be COOL and COLD; Inference: Rules 1 and 2 will fire: Composition: The temperature is COLD with a truth value of µ COLD=0.5 the SPEED will be k1 The temperature is COOL with a truth value of µcool =0.5 the SPEED will be k2 DEFUZZIFICATION : CONTROL speed is (µ COLD*k1+ µcool *k2)/(µ COLD+ µcool)= (0.5*0+0.5*30)/( )=15 RPM 58 58

59 Knowledge Representation & Reasoning: The Air-conditioner Example Zero Order Takagi Sugeno Controller Membership Function Speed MINIMAL SLOW MEDIUM FAST BLAST 59 59

60 Knowledge Representation & Reasoning: The Air-conditioner Example DEFUZZIFICATION: Comparing the results of two model identification exercises Mamdani and Takagi-Sugeno- we get the following results: Controller Centre of Area Takagi- Sugeno (RPM) Mamdani (RPM) Mean of Maxima

61 Neuro-fuzzy models Learn from the input-output data: Data mining; Machine Learning; Neural Networks; } Soft Computing Genetic Algorithms Hybrids Neuro Fuzzy systems Jang, Jyh-Shing Roger., Sun, Chuen-Tsai & Mizutani, Eiji. (1997). Neuro-Fuzzy & Soft Computing: A Computational Approach to Learning and Machine Intelligence. Upper Saddle River (NJ): Prentice Hall, Inc. (Chapters 8 and 12) 61 61

62 Neuro-fuzzy models Learn from the input-output data: If a soft computing system is able to compute the input-output relationships, then it will LEARN to compute the relationships Jang, Jyh-Shing Roger., Sun, Chuen-Tsai & Mizutani, Eiji. (1997). Neuro-Fuzzy & Soft Computing: A Computational Approach to Learning and Machine Intelligence. Upper Saddle River (NJ): Prentice Hall, Inc. (Chapters 8 and 12) 62 62

63 Neuro-fuzzy models: A case study Consider a first-order Sugeno fuzzy model with two inputs (x & y) and one output (z). There are two fuzzy rules: R1: IF x is A 1 and y is B 1 THEN f 1 =p 1 x+q 1 y+r 1 R2: IF x is A 2 and y is B 2 THEN f 2 =p 2 x+q 2 y+r

64 Neuro-fuzzy models: A case study Consider a first-order Sugeno fuzzy model with two inputs (x & y) and one output (z). Layer 1 Layer 2 x A 1 TT w 1 N w 1 w 1 f 1 Layer 5 A 2 f y B 1 TT w 2 N w 2 w 2 f 2 B 2 Layer 3 Layer

65 Neuro-fuzzy models: A case study The operation of a fuzzy system depends on the execution of FOUR major tasks: Fuzzification, Inference, Composition, Defuzzification. The different layers in an adaptive network perform one or more of the tasks 65 65

66 Neuro-fuzzy models: A case study Consider a first-order Sugeno fuzzy model with two inputs (x & y) and one output (z). LAYER 5: The single node in this layer is a fixed node labelled, which computes the overall output as the summation of all incoming signals O 5,1 = w f = i _ i i i i w i w f i i 66 66

67 Neuro-fuzzy models: A case study The network below is an adaptive network that is functionally equivalent to a Takagi-Sugeno model. Layer 1 Layer 2 x A 1 TT w 1 N w 1 w 1 f 1 Layer 5 A 2 f y B 1 TT w 2 N w 2 w 2 f 2 B 2 Layer 3 Layer

68 Notes on Artificial Neural Networks: The fan-ins and fan-outs neurons with 10 4 connections and an average of 10 spikes per second = 1015 adds/sec. This is a lower bound on the equivalent computational power of the brain fan-in Asynchronous firing rate, c. 200 per sec. summation 4 10 fan-out meters per sec

69 Notes on Artificial Neural Networks: Biological and Artificial NN s Entity Processing Units Input Output Biological Neural Networks Neurons Dendrites Axons Artificial Neural Networks Network Nodes Network Arcs Network Arcs Inter-linkage Synaptic Contact (Chemical and Electrical) Plastic Connections Node to Node via Arcs Weighted Connections Matrix 69 69

70 Notes on Artificial Neural Networks: Rosenblatt s Perceptron A single layer perceptron can carry out a number can perform a number of logical operations which are performed by a number of computational devices. A learning perceptron below performs the AND operation. An algorithm: Train the network for a number of epochs (1) Set initial weights w1 and w2 and the threshold θ to set of random numbers; (2) Compute the weighted sum: x 1 *w 1 +x 2 *w 2 + θ (3) Calculate the output using a delta function y(i)= delta(x 1 *w 1 +x 2 *w 2 + θ ); delta(x)=1, if x is greater than zero, delta(x)=0,if x is less than equal to zero (4) compute the difference between the actual output and desired output: e(i)= y(i)-y desired (5) If the errors during a training epoch are all zero then stop otherwise update w j (i+1)=w j (i)+ α*x j *e(i), j=1,

71 Neuro-fuzzy models Adaptive Networks A network typically comprises a set of nodes connected by directed links. Each node performs a static node function on its incoming signals to generate a single node output. Each link specifies the direction of signal flow from one node to another. An adaptive network is a network structure whose overall input-output behaviour is determined by a collection of modifiable parameters

72 Neuro-fuzzy models Neural Networks 'learn' by adapting in accordance with a training regimen: The network is subjected to particular information environments on a particular schedule to achieve the desired end-result. There are three major types of training regimens or learning paradigms: SUPERVISED UN-SUPERVISED REINFORCEMENT or GRADED 72 72

73 Supervised Learning Our focus will be on supervised learning, particularly networks that learn by using the so-called back-propagation algorithm and comprise hidden layers between the input & the output layers

74 Supervised Learning A situation in which the network is functioning as an input/output system. The network receives a vector v and emits another, v. Supervised learning regimen involves the network being supplied with a sequence of examples v v v ( ) ( v ) v ( ) of "desireable" or "correct" input/output pairs. For each input v the network is supplied v, the correct output

75 Other Learning Systems UNSUPERVISED LEARNING or SELF-ORGANISATION: Under this regimen a network modifies itself in response to v inputs. There are no v inputs or a grade/score (see next page). Therefore, in unsupervised or self-organisation learning there is no EXTERNAL TEACHER or CRITIC to oversee the learning process. ENVIRONMENT vector describing state of the environment TEACHER In an unsupervised regimen there are no specific examples of the function to be learned by the network

76 Rosenblatt s Perceptrons A perceptron computes a binary function of its input. A group of perceptrons can be trained on sample input-output pairs until it learns to compute the correct function. Each perceptron, in some model, can function independently of others in the group, they can be separately trained linearly separable. Thresholds can be varied together with weights. Given values of x 1 and x 2 to train such that the perceptron outputs 1 for white dots and 0 for black dots

77 Back-propagation Algorithm: Supervised Learning Backpropagation (BP) is amongst the most popular algorithms for ANNs : it has been estimated by Paul Werbos, the person who first worked on the algorithm in the 1970 s, that between 40% and 90% of the real world ANN applications use the BP algorithm. Werbos traces the algorithm to the psychologist Sigmund Freud s theory of psychodynamics. Werbos applied the algorithm in political forecasting. David Rumelhart, Geoffery Hinton and others applied the BP algorithm in the 1980 s to problems related to supervised learning, particularly pattern recognition. The most useful example of the BP algorithm has been in dealing with problems related to prediction and control

78 Back-propagation Algorithm : Worked Example #1 Consider a 2x2x1 network: The desired vector d=0.9 First Layer Connectivity w w w w Bias weight 0.2 = [ w w w ] [ ] = Second Layer Connectivity [ w w ] [ ] =

79 Back-propagation Algorithm : Worked Example #1 Consider a 2x2x1 network: d= Consider an input vector x: x x = [ ] [ ] 79 79

80 Back-propagation Algorithm : Worked Example #1 Consider a 2x2x1 network: d=